Existence and uniqueness of neutral functional differential equations with sequential fractional operators

In this research paper, we investigate the existence and uniqueness of solutions for neutral functional differential equations with sequential fractional orders, specifically involving the G-Caputo operator. To obtain the desired results, we employ the Banach fixed point theorem (BFPT), a nonlinear variation of the Leray-Schauder fixed point theorem (SFPT), and the Krasnoselski fixed point theorem (KFPT). Additionally, we provide illustrative examples that demonstrate the key findings. Furthermore, we address a scenario where an initial value integral condition is considered.

Let u be defined on ½a À t; b�: Then for r 2 J, we denote by u r 2 C t ≔ Cð½À t; 0�; RÞ given by u r ðyÞ ¼ uðr þ yÞ; y 2 ½À t; 0�: Differential functional equations play a crucial role across various domains, including control theory, neural networks, and epidemiology [2].Specifically, delay differential equations prove invaluable in understanding the dynamics of real populations, handling both finite and infinite delay terms.The presence of delays in derivatives poses challenges, prompting an alternative approach through the exploration of neutral functional differential equations.Additionally, fractional derivatives become instrumental in capturing hereditary as well as memory effects in diverse materials and processes.Thus, making the investigation of functional neutral differential equations that involve fractional derivatives a vital area of research.Hybrid problems, which involve the combination of different mathematical techniques or models to solve complex physical phenomena, have applications in various fields.Some examples of physical applications of hybrid problems include Fluid-structure interaction, electromagnetic-structural interaction, thermo-mechanical analysis, coupled heat and mass transfer, multiphase flow with chemical reactions, biomechanics.Comprehensive information on this topic can be found in referenced texts [3][4][5][6][7].
In a previous study [27][28][29][30], researchers focused on an IVP involving fractional functional and neutral functional differential equations with infinite delay of the Riemann-Liouville type.Subsequent investigations [31] extended to IVPs concerning functional and neutral functional differential equations, including fractional order of Hadamard type.Another study [32] addressed an initial value problem involving retarded functional Caputo-type fractional impulsive differential equations with variable moments.
Fixed-point theory, a mathematical branch, explores the existence and properties of fixed points in mappings or functions.With applications across mathematics and various fields like economics, computer science, physics, engineering, and social sciences, fixed-point theory provides a framework for studying solutions to equations and mappings.The current research focuses on a novel class of sequential fractional neutral functional differential equations, particularly of the G-Caputo type.Utilizing fixed point theorems by Banach and Krasnoselskii [33] together with the nonlinear version of Leray-Schauder type introduced by [34], this study advances our understanding of these equations.
The paper's remaining sections are organized as follows: Section 2 reviews crucial preliminaries for subsequent analysis.Section 3 examines the existence and uniqueness of solutions for the presented problem in Eqs (1) and (2).Section 4 presents the existence results for the problem.Section 5 introduces a generalization incorporating an initial value integral condition, along with illustrative examples showcasing the study's findings.Finally, Section 6 constructs additional illustrative examples to reinforce the obtained results.

Essentiel preliminaries
In the following section, we will introduce the notation, definitions, and preliminary facts that are essential for the subsequent analysis.We denote by CðJ; RÞ the Banach space consisting of all continuous functions from the interval J to the real numbers R. The norm associated with this space is given by kuk 1 ≔ supfjuðrÞj : r 2 Jg: Let G : ½a; I� !R be increasing and G 0 ðrÞ 6 ¼ 0; 8r.Definition 1.1 ([8, 35]).The G-Riemann-Liouville fractional integral (G-RLFI) of order α > 0 for a CF u : ½a; I� !R is referred to as 8,35]).The G-Caputo fractional derivative (G-CFD) of order α > 0 for a CF u : ½a; I� !R is the aim of where 8,35]).Let q, ℓ > 0, and u 2 Cð½a; b�; RÞ.Then, 8r 2 ½a; b�, and by assuming F a ðrÞ ¼ GðrÞ À GðaÞ; we have Besides, and Lemma 1.5.The function u 2 C 2 ð½a À t; b�; RÞ is a solution of the problem Proof.The solution of G-Caputo differential Eq in ( 4) can be written as upon considering the condition D a 2 ;G a uðaÞ ¼ Z, with s 1 2 R being an arbitrary constant, we determine that σ 1 can be expressed as s 1 ¼ Z À ℏða; �ðaÞÞ.Subsequently, we derive the following result: The equation above allows us to determine that s 2 ¼ �ðaÞ and the validity of ( 5) is demonstrated.The reverse can be deduced through straightforward calculations.
The following theorem provides a uniqueness outcome based on the given assumptions.

Existence results
This section is devoted to the presentation of the findings regarding the existence of solutions for the initial value problem outlined in Eqs ( 1)-( 3).The next result is based on the nonlinear version of the Leray-Schauder theorem.Lemma 3.1 ([34]).Consider E as a Banach space, C as a closed and convex subset of C, and U as an open subset of C with the inclusion of 0 2 U. Assuming that R is a continuous and compact mapping (meaning Rð � U Þ is a subset of C that is relatively compact), then either: (ii) there is a X 2 @U (the boundary of U in C) and λ 2 (0, 1) with X ¼ lRðXÞ.
Step 1: @ is continuously defined.Consider a sequence fu n g such that u n !u in Cð½a À t; b�; RÞ.So j@ðu n ÞðrÞ À @ðuÞðrÞj Because both F and ℏ are continuous functions, we obtain Step 2: The operator @ transforms bounded sets into bounded sets within the function space Cð½a À t; b�; RÞ.Sufficiently to prove for any θ > 0 there exists ' > 0 such that for each u 2 Q y ¼ fu 2 Cð½a À t; b�; RÞ : kuk 1 � yg, we have k@ðuÞk 1 � '.By (O4) and (O5), for each r 2 J, we get ðF a ðbÞÞ a 1 þa 2 ≔ ': Step 3: Now we show that the image of a bounded set by @ is an equicontinuous subset of Cð½a À t; b�; RÞ.Indeed; suppose that r 1 , r 2 2 J, r 1 < r 2 , Q y is a bounded subset of Cð½a À t; b�; RÞ, and take u 2 Q y .We have As r 1 approaches r 2 , the right-hand side of the mentioned inequality approaches zero.Equicontinuity in cases where r 1 is less than r 2 and when r 1 is less than or equal to zero while r 2 is greater than or equal to zero is readily apparent.
As a result of Steps 1 to 3, applying the theorem of Arzela ´-Ascoli leads to the fact that @ : Cð½a À t; b�; RÞ !Cð½a À t; b�; RÞ, is continuous as well as completely continuous.
Step 4: We demonstrate the existence of an open set U � Cð½a À t; b�; RÞ with u 6 ¼ l@ðuÞ for λ 2 (0, 1) and u 2 @U.Let u 2 Cð½a À t; b�; RÞ and u ¼ l@ðuÞ for some 0 < λ < 1.Then, for each r 2 J, we have Based on the assumptions we have made, for every r 2 J, we achieve we can also write it as follows In view of (O6), there exists M such that kuk ½aÀ t;b� 6 ¼ M. Let us set U ¼ fu 2 Cð½a À t; b�; RÞ : kuk ½aÀ t;b� < Mg: It's worth noting that the operator @ : � U ! Cð½a À t; b�; RÞ, exhibits both continuity and complete continuity.Given the selection of U, there exists no u on the boundary @U such that u ¼ l@u for any λ in (0, 1).Consequently, employing the nonlinear alternative of Leray-Schauder type (Lemma 3.1), we conclude that @ possesses a fixed point u 2 � U , which serves as a solution to the problem described in (1)-( 3).This successfully concludes the proof.Lemma 3.3 (KFPT [33]).Consider S as a closed, bounded, convex, and non-empty subset of a Banach space X.Operators P and Q are defined as follows: • Py þ Qx 2 S whenever y; u 2 S; • P is continuous and compact; Then there exists z 2 S so that z ¼ Pz þ Qz.Theorem 3.4.Assume that the conditions (O2) and (O3) are satisfied and that (O7) jF ðr; xÞj � mðrÞ, jℏðr; xÞj � nðrÞ, for all ðr; xÞ 2 J � R, and m; n 2 CðJ; R þ Þ.Then we have at least one solution of problem (1)-(3) on ½a À t; b�, if kðF a ðbÞÞ Proof.We define the operators X 1 and X 2 by Setting sup r2½a;b� mðrÞ ¼ kmk 1 ; sup r2½a;b� nðrÞ ¼ knk 1 and choosing we consider Q r ¼ fu 2 Cð½a À t; b�; RÞ : kuk 1 � rg.For any u; z 2 Q r , we have Since F is continuous, the operator X 2 is continuous.Additionally, X 2 is uniformly bounded on Q r by Next, we will establish the compactness of the operator X 2 .To do this, we introduce the following definitions F ¼ sup ðr;uÞ2½a;b��Q r jF ðr; uÞj < 1; and consequently, for r 1 ; r 2 2 ½a; b�; r 1 < r 2 , we have which is independent of u and tends to zero as r 2 − r 1 !0. Thus, X 2 is equicontinuous.So X 2 is relatively compact on Q r .Hence, by the Arzelá-Ascoli theorem, X 2 is compact on Q r .Thus all the assumptions of Lemma 3.3 are satisfied.So the conclusion of Lemma 3.3 implies that the problem (1)-( 3) has at least one solution on ½a À t; b�

Initial value integral
The findings presented in this work can be expanded to encompass scenarios involving an initial value integral condition structured as follows: In the context where W : J � Cð½À t; 0�; RÞ !R, the variable η within Eq (8) will be substituted with R b a Wðs; u s Þds.As a consequence, we can express the existence and uniqueness statement for the problem encompassing Eqs (1) and ( 2)-( 13) in the following manner.
Theorem 4.1.Given the fulfillment of conditions (O1) and (O2), we also make the additional assumption that (O8) there exists a nonnegative constant m such that jWðr; XÞ À Wðr; YÞj � mkX À Yk C ; for r 2 J and any X; Y 2 C t : Hence the problem (1) and ( 2)-( 13) has a unique solution on ½a À t; b� if The proof of the aforementioned theorem closely resembles that of Theorem 2.2.The analogous form of the existence results, as seen in Theorems 3.2 and 3.4, can be developed for the problem described by Eqs (1) and ( 2)-( 13) in a comparable fashion.

Example
In this section, we provide an example to demonstrate how our primary findings can be applied effectively.We will examine a fractional functional differential equation to illustrate this concept, let GðrÞ ¼ log r   Gða 2 þ1Þ þ 'ðF 1 ðbÞÞ a 1 þa 2 Gða 1 þa 2 þ1Þ � 0:910079666 < 1, hence as asserted by Theorem 2.2, the problem encompassing Eqs ( 14 Gða 2 þ1Þ � 0:487071271 < 1.The conditions stated in Theorem 3.4 are evidently met.As a direct consequence of the theorem's conclusion, a solution to the problem described in Eqs ( 14)-( 16) is guaranteed to exist within the interval [1 − τ, e].

Conclusion
In our research, we touched on the theory of existence and uniqueness of a kind of complex equations included neutral functional differential equations by attracting the generalized fractional derivation G-Caputo derivative.